Using the weak heavens condition we find a non-linear partial differential equation which is shown to generalize the heavenly equation of Self-dual gravity. This differential equation we call “weak heavenly equation” (\({\mathcal {WH}}\)-equation). For the two-dimensional case the \({\mathcal {WH}}\)-equation is brought into the evolution (Cauchy–Kovalevski) form using a Legendre transformation. Finally, we find that this transformed equation (“evolution weak heavenly equation”) does admit very simple solutions.

A massless scalar field in the \((1 + n)\)-dimensional Minkowski spacetime is considered. The d’Alembert equation for the field is solved in pseudospherical coordinates. The positive frequency part of the solutions is found using the symmetry between coordinates and momenta. The solutions are projected onto the \(n\)-dimensional de Sitter hyperboloid em- bedded in the fiat spacetime. The results for the special \(n=4\) case are easily reproduced.

It is shown that the simple consecutive scattering scheme with strangeness yield in nucleon-nucleon pair collision dependent on the “composition” of the pair (fresh–fresh, wounded–fresh or wounded–wounded) can reproduce some aspects of strangeness enhancement observed in relativistic nuclear interactions.

Electron-nucleus elastic scattering is calculated by means of the multiple-scattering theory of Glauber and the eikonal approximation applied to different forms of nuclear charge distribution. The differences between the calculated cross-sections are examined and compared with the experimental data for \(^{12}\)C, \(^{16}\)Q, \(^{40}\)Ca and \(^{208}\)Pb nuclei at different energies. A very good agreement with the experimental data, at higher angles, is obtained for densities other than Gaussian. The multiple-scattering shows even better agreement than the eikonal approach for lighter nuclei, especially at the diffraction dip.

Deuteron spectra from the \(^{60}\)Ni\((n,d)\)\(^{59}\)Co reaction has been measured at the reaction angles \(0^\circ \)–\(70^\circ \) with an eight-telescope setup. Angular distribution for transitions to the ground state and to groups of excited states were obtained. Distorted wave Born approximation analysis of the angular distributions gives spectroscopic factors of 5.8, 0.71, 3.37, 0.9, 7.4 for the ground and excited states, respectively.

To resolve infrared problems with the de Sitter invariant vacuum, de Sitter symmetry must be broken. We discuss how the history of the de Sitter phase is related to the infrared cutoff. We illustrate how either (1) the diagonalisation of the Hamiltonian for long-wavelength modes or (2) an explicit modification of the metric in the distant past leads to natural infrared cutoffs. The former case resembles a bosonic superconductor in which graviton-pairing occurs between non-adiabatic modes. While the dynamical equations respect de Sitter symmetry, the vacuum is not de Sitter invariant because of the introduction of an initial condition at a finite time. In the latter case, we indicate just how infrared cutoffs are associated with modifications of the metric. The implications for the one-loop stress tensor and the production of particles are also discussed.